Time-dependent Partial Differential Equations Research Articles

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

GRINN: a physics-informed neural network for solving hydrodynamic systems in the presence of self-gravity

Modeling self-gravitating gas flows is essential to answering many fundamental questions in astrophysics. This spans many topics including planet-forming disks, star-forming clouds, galaxy formation, and the development of large-scale structures in the Universe. However, the nonlinear interaction between gravity and fluid dynamics offers a formidable challenge to solving the resulting time-dependent partial differential equations (PDEs) in three dimensions (3D). By leveraging the universal approximation capabilities of a neural network within a mesh-free framework, physics informed neural networks (PINNs) offer a new way of addressing this challenge. We introduce the gravity-informed neural network (GRINN), a PINN-based code, to simulate 3D self-gravitating hydrodynamic systems. Here, we specifically study gravitational instability and wave propagation in an isothermal gas. Our results match a linear analytic solution to within 1% in the linear regime and a conventional grid code solution to within 5% as the disturbance grows into the nonlinear regime. We find that the computation time of the GRINN does not scale with the number of dimensions. This is in contrast to the scaling of the grid-based code for the hydrodynamic and self-gravity calculations as the number of dimensions is increased. Our results show that the GRINN computation time is longer than the grid code in one- and two- dimensional calculations but is an order of magnitude lesser than the grid code in 3D with similar accuracy. Physics-informed neural networks like GRINN thus show promise for advancing our ability to model 3D astrophysical flows.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

Positional embeddings for solving PDEs with evolutional deep neural networks

This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity.Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach

It is suggested that an explicit-implicit third-order in-time numerical scheme be used to solve time-dependent partial differential equations (PDEs). A fourth-order compact scheme in space is adopted for discretizing space variables. The work creatively blends a third-order explicit-implicit numerical scheme in time with a fourth-order compact scheme in space to solve time-dependent PDEs. Notably, the proposed numerical scheme demonstrates stability for scalar PDEs and exhibits convergence for both linear and nonlinear PDE systems. In addition, a mathematical model is given for the heat transfer of magnetohydrodynamic (MHD) flow over flat and oscillatory sheets, and the effect of viscous dissipation is also considered. For various governing parameters, features of the flow and heat transfer characteristics are discussed and examined. Some results are verified using an exact solution. We use an explicit-implicit finite difference method guaranteed to be stable and convergent under all conditions to solve the coupled nonlinear partial differential equations that describe this problem. Physical aspects of the problem are discussed in light of the system’s embedded parameters, and numerical and graphical results are presented. To sum up, progress in our knowledge of such systems relies heavily on creating an accurate and efficient numerical scheme for modeling MHD boundary layer flows while considering the influence of viscous dissipation. This study aims to improve the current state of MHD flow simulations and provide a useful resource for engineers and scientists engaged in developing and improving MHD-based technologies.

Neural Oscillators for Generalization of Physics-Informed Machine Learning

A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

Neural Physical Simulation with Multi-Resolution Hash Grid Encoding

We explore the generalization of the implicit representation in the physical simulation task. Traditional time-dependent partial differential equations (PDEs) solvers for physical simulation often adopt the grid or mesh for spatial discretization, which is memory-consuming for high resolution and lack of adaptivity. Many implicit representations like local extreme machine or Siren are proposed but they are still too compact to suffer from limited accuracy in handling local details and a long time of convergence. We contribute a neural simulation framework based on multi-resolution hash grid representation to introduce hierarchical consideration of global and local information, simultaneously. Furthermore, we propose two key strategies: 1) a numerical gradient method for computing high-order derivatives with boundary conditions; 2) a range analysis sample method for fast neural geometry boundary sampling with dynamic topologies. Our method shows much higher accuracy and strong flexibility for various simulation problems: e.g., large elastic deformations, complex fluid dynamics, and multi-scale phenomena which remain challenging for existing neural physical solvers.

Numerical investigation of high-dimensional option pricing PDEs by utilizing a hybrid radial basis function - finite difference procedure

The target of this research is to resolve high-dimensional partial differential equations (PDEs) for multi-asset options, modeled as parabolic time-dependent PDEs. We present a hybrid radial basis function - finite difference (RBF-FD) solver, which combines the advantages of Gaussian and multiquadric functions. Additionally, we employ the Krylov subspace method on the resulting system of ordinary differential equations, reducing the computation load for finding the numerical solution. Computational tests, up to seven dimensions, and comparisons support the superiority of our hybrid solver.

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Proper orthogonal decomposition method of constructing a reduced-order model for solving partial differential equations with parametrized initial values

This paper proposes a novel proper orthogonal decomposition method for constructing a reduced-order model. This model effectively computes solutions for various initial conditions in time-dependent partial differential equations. The mode obtained from the proposed proper orthogonal decomposition captures both the fluctuating component and the mean-field of the time-dependent solution. In this mode, the eigenvalues representing the mean-field, are significantly larger than those representing the fluctuating component. Consequently, determining the number of modes required for representing a solution cannot rely solely on the cumulative contribution rate. Therefore, the proposed method gives a scalar weight to the mean-field and controls the magnitude of the mean-field energy. We propose a method that considers the magnitude of the scaler weight alongside the cumulative contribution ratio for determining the number of modes. This method was evaluated using the time-dependent Burgers equation with parametric initial values. Our proposed method selects the appropriate mode to represent the given dataset, unlike the conventional method. A reduced-order model based on the proposed method effectively computes the time evolutions of the solutions using datasets for several parameters by imposing a condition that the cumulative contribution ratio exceeds 98%.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn–Hilliard, Korteweg–de Vries, Allen–Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

A finite difference explicit-implicit scheme for fractal heat and mass transportation of Williamson nanofluid flow in quantum calculus

This study introduces a novel and versatile fractal finite difference scheme designed to address unsteady flow challenges in quantum calculus over flat and oscillatory sheets. Motivated by the need to advance our understanding of nanofluid dynamics, particularly in heat and mass transportation, this research aims to bridge existing gaps and contribute to improving computational methodologies. Our approach involves a two-stage scheme incorporating explicit and implicit methods, focusing on q-derivatives for spatial terms within the mathematical model. The system encompasses the Navier-Stokes equation, energy equation, and concentration equation, offering a comprehensive framework for analyzing mixed convective Williamson nanofluid flow under the influence of chemical reactions. The convergence condition for the time-dependent partial differential equations system, particularly the scalar convection-diffusion equation, is presented. Stability requirements for our proposed fractal scheme are also delineated. Notably, our study explores the implications of the Weisenberg parameter on the velocity profile, revealing a decline that underscores its influence on nanofluid behavior. Our fractal finite difference scheme demonstrates a substantially faster convergence rate than the fractal Crank-Nicolson scheme. This outcome emphasizes the computational efficiency and efficacy of our proposed technique. In a broader context, our work contributes to the field by leveraging fractal geometry and quantum calculus to enhance our understanding of nanofluid dynamics. These tools present exciting opportunities for groundbreaking applications and technological advancements across various industries. As we delve into these research areas, our work sets the stage for exploration, providing a framework with great potential for transformative progress in nanofluid dynamics.

Energy-preserving schemes for conservative PDEs based on periodic quasi-interpolation methods

In this paper, we present a novel energy-preserving scheme for solving time-dependent partial differential equations with periodic solutions on non-uniform grids. The proposed scheme combines the periodic quasi-interpolation approaches with the discrete gradient method. One of the main advantages of our method is its ability to ensure energy conservation without relying on the requirement of an anti-symmetric differentiation matrix. We rigorously verify the energy conservation properties of the full-discrete scheme, as well as establish the convergence results. Furthermore, we extend the capabilities of our method by incorporating an adaptive strategy, allowing for the design of an adaptive energy-preserving scheme. This adaptive feature enhances the accuracy and efficiency of the numerical solution. To demonstrate the effectiveness and energy-preserving ability of the proposed scheme, we present several numerical examples. These examples showcase the superior accuracy and robustness of our method.

Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.

物理信息编码解码卷积长短期记忆网络求解时变偏微分方程

物理信息编码解码卷积长短期记忆网络求解时变偏微分方程

Magneto-thermal convection and entropy production of hybrid nanofluid in an inclined chamber having a solid block

PurposeThis paper aims to examine the thermal transmission and entropy generation of hybrid nanofluid filled containers with solid body inside. The solid body is seen as being both isothermal and capable of producing heat. A time-dependent non-linear partial differential equation is used to represent the transfer of heat through a solid body. The current study’s objective is to investigate the key properties of nanoparticles, external forces and particular attention paid to the impact of hybrid nanoparticles on entropy formation. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates. Hybrid nanofluid has been proven to have useful qualities, making it an attractive coolant for an electrical device. The findings would help scientists and engineers better understand how to analyse convective heat transmission and how to forecast better heat transfer rates in cutting-edge technological systems used in industries such as heat transportation, power generation, chemical production and passive cooling systems for electronic devices.Design/methodology/approachThermal transmission and entropy generation of hybrid nanofluid are analysed within the enclosure. The domain of interest is a square chamber of size L, including a square solid block. The solid body is considered to be isothermal and generating heat. The flow driven by temperature gradient in the cavity is two-dimensional. The governing equations, formulated in dimensionless primitive variables with corresponding initial and boundary conditions, are worked out by using the finite volume technique with the SIMPLE algorithm on a uniformly staggered mesh. QUICK and central difference schemes were used to handle convective and diffusive elements. In-house code is developed using FORTRAN programming to visualize the isotherms, streamlines, heatlines and entropy contours, which are handled by Tecplot software. The influence of nanoparticles volume fraction, heat generation factor, external magnetic forces and an irreversibility ratio on energy transport and flow patterns is examined.FindingsThe results show that the hybrid nanoparticles concentration augments the thermal transmission and the entropy production increases also while the augmentation of temperature difference results in a diminution of entropy production. Finally, magnetic force has the significant impact on heat transfer, isotherms, streamlines and entropy. It has been observed that the external magnetic force plays a good role in thermal regulations.Research limitations/implicationsHybrid nanofluid is a desirable coolant for an electrical device. Various nanoparticles and their combinations can be analysed. Ferro-copper hybrid nanofluid considered with the help of prevailing literature review. The research would benefit scientists and engineers by improving their comprehension of how to analyses convective heat transmission and forecast more accurate heat transfer rates in various fields.Practical implicationsDue to its helpful characteristics, ferrous-copper hybrid nanofluid is a desirable coolant for an electrical device. The research would benefit scientists and engineers by improving their comprehension of how to analyse convective heat transmission and forecast more accurate heat transfer rates in cutting-edge technological systems used in sectors like thermal transportation, cooling systems for electronic devices, etc.Social implicationsEntropy generation is used for an evaluation of the system’s performance, which is an indicator of optimal design. Hence, in recent times, it does a good engineering sense to draw attention to irreversibility under magnetic force, and it has an indispensable impact on investigation of electronic devices.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyse convective energy transport and entropy generation in a chamber with internal block, which is capable of maintaining heat and producing heat. Effects of irreversibility ratio are scrutinized for the first time. Analysis of convective heat transfer and entropy production in an enclosure with internal isothermal/heat generating blocks gives the way to predict enhanced heat transfer rate and avoid the failure of advanced technical systems in industrial sectors.